A singularly perturbed nonlinear Robin problem in a periodically perforated domain: A functional analytic approach

Massimo Lanza de Cristoforis*, Paolo Musolino

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Citations (SciVal)

Abstract

Let n ∈ ℕ\{0, 1}. Let q be the n × n diagonal matrix with entries q11,..., qnn in] 0, +∞[. Then qℤn is a q-periodic lattice in ℝn with fundamental cell Q ≡ Πn j=0]0, qjj[. Let p ∈ Q. Let Ω be a bounded open subset of ℝn containing 0. Let G be a (nonlinear) map from ∂Ω × ℝ to ℝ. Let γ be a positive-valued function defined on a right neighbourhood of 0 in the real line. Then we consider the problem for ε > 0 small, where νp+εΩ denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition limε→0+γ(ε)-1ε ∈ ℝ, we prove that the above problem has a family of solutions {u(ε, ·)}ε∈]0, ε′[ for ε′ sufficiently small, and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis.

Original languageEnglish
Pages (from-to)511-536
Number of pages26
JournalComplex Variables and Elliptic Equations
Volume58
Issue number4
Early online date10 Jan 2012
DOIs
Publication statusPublished - 01 Apr 2013

Keywords

  • Laplace operator
  • periodic nonlinear Robin boundary-value problem
  • real-analytic continuation in Banach space
  • singularly perturbed data
  • singularly perturbed domain

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